Question

activity 4 (continued)

1. each of these triangles is labelled with a letter.

describe, as an ordered pair, the translations which maps
a) a onto c

b) c onto a

c) d onto b

d) b onto d

e) c onto d

f) d onto c

summary of translations
→ with a translation, although the position of the shape has changed, its size and shape are the same before and after the translation.
→ for translations of a point or shape to the left or to the right (horizontally) but not up or down (vertically), the x-value of the coordinate changes and the y-value stays the same.
→ for translations of a point or shape up or down (vertically) but not to the left or the right (horizontally), the y-value changes and the x-value stays the same.
→ a translation of a figure (the object) produces another figure (the image) that is congruent to the first figure

Explanation:

To solve the translation problems, we analyze the horizontal (x - direction) and vertical (y - direction) shifts between the triangles. Let's assume each grid square represents 1 unit.

Part (a): Map A onto C

• Step 1: Determine vertical shift

Triangle A is above triangle C. To move from A to C, we move down. Count the number of grid units: A is 2 units above C, so the vertical change (Δy) is - 2 (downward is negative in the y - direction).

• Step 2: Determine horizontal shift

Triangle A and C are aligned horizontally (no left/right shift). So the horizontal change (Δx) is 0.
The translation vector is \((0, - 2)\).

Part (b): Map C onto A

• Step 1: Determine vertical shift

Triangle C is below triangle A. To move from C to A, we move up. The vertical change (Δy) is + 2 (upward is positive in the y - direction).

• Step 2: Determine horizontal shift

No horizontal shift (Δx = 0).
The translation vector is \((0, 2)\).

Part (c): Map D onto B

• Step 1: Determine vertical shift

Triangle D is below triangle B. To move from D to B, we move up. Count the units: D is 3 units below B, so Δy = + 3.

• Step 2: Determine horizontal shift

Triangle D is to the right of triangle B. To move from D to B, we move left. Count the units: D is 3 units to the right of B, so Δx = - 3.
The translation vector is \((- 3, 3)\).

Part (d): Map B onto D

• Step 1: Determine vertical shift

Triangle B is above triangle D. To move from B to D, we move down. Δy = - 3.

• Step 2: Determine horizontal shift

Triangle B is to the left of triangle D. To move from B to D, we move right. Δx = + 3.
The translation vector is \((3, - 3)\).

Part (e): Map C onto D

• Step 1: Determine vertical shift

Triangle C is above triangle D. To move from C to D, we move down. Δy = - 1 (assuming the vertical distance is 1 unit? Wait, re - evaluating: Wait, looking at the grid, C and D: Let's check the vertical and horizontal. Wait, C is to the left of D and above? Wait, no, let's re - check. Wait, maybe my initial count was wrong. Wait, let's take a vertex of C and D. Let's say the top - left vertex of C and D. For C to D: horizontal shift: D is 6 units to the right of C? Wait, no, maybe the grid is such that from C to D: horizontal shift (right) is 6? Wait, no, the original figure: Let's assume each triangle has a top - left vertex. For A: top - left at (x1,y1), B: (x1 + 5,y1), C: (x1,y1 - 2), D: (x1+6,y1 - 3). Wait, maybe better to count the horizontal and vertical differences. Let's take the top - left corner of each triangle:

• A: (let's say) (2, 5)
• B: (7, 5)
• C: (2, 3)
• D: (8, 2)

So for C (2,3) to D (8,2):

• Horizontal shift: 8 - 2 = 6 (right, so + 6)
• Vertical shift: 2 - 3 = - 1 (down, so - 1)

So translation vector is \((6, - 1)\). Wait, maybe my initial grid assumption was wrong. Alternatively, maybe the horizontal shift from C to D is 6 units right and 1 unit down.

Part (f): Map D onto C

• Step 1: Determine vertical shift

Triangle D is below triangle C. To move from D to C, we move up. Δy = + 1.

• Step 2: Determine horizontal shift

Triangle D is to the right of triangle C. To move from D to C, we move left. Δx = - 6.
The translation vector is \((- 6, 1)\).

Final Answers:

a) \(\boldsymbol{(0, - 2)}\)
b) \(\boldsymbol{(0, 2)}\)
c) \(\boldsymbol{(- 3, 3)}\)
d) \(\boldsymbol{(3, - 3)}\)
e) \(\boldsymbol{(6, - 1)}\) (depending on grid interpretation, may vary slightly but the method is horizontal and vertical shift calculation)
f) \(\boldsymbol{(- 6, 1)}\)

Answer:

To solve the translation problems, we analyze the horizontal (x - direction) and vertical (y - direction) shifts between the triangles. Let's assume each grid square represents 1 unit.

Part (a): Map A onto C

• Step 1: Determine vertical shift

Triangle A is above triangle C. To move from A to C, we move down. Count the number of grid units: A is 2 units above C, so the vertical change (Δy) is - 2 (downward is negative in the y - direction).

• Step 2: Determine horizontal shift

Triangle A and C are aligned horizontally (no left/right shift). So the horizontal change (Δx) is 0.
The translation vector is \((0, - 2)\).

Part (b): Map C onto A

• Step 1: Determine vertical shift

Triangle C is below triangle A. To move from C to A, we move up. The vertical change (Δy) is + 2 (upward is positive in the y - direction).

• Step 2: Determine horizontal shift

No horizontal shift (Δx = 0).
The translation vector is \((0, 2)\).

Part (c): Map D onto B

• Step 1: Determine vertical shift

Triangle D is below triangle B. To move from D to B, we move up. Count the units: D is 3 units below B, so Δy = + 3.

• Step 2: Determine horizontal shift

Triangle D is to the right of triangle B. To move from D to B, we move left. Count the units: D is 3 units to the right of B, so Δx = - 3.
The translation vector is \((- 3, 3)\).

Part (d): Map B onto D

• Step 1: Determine vertical shift

Triangle B is above triangle D. To move from B to D, we move down. Δy = - 3.

• Step 2: Determine horizontal shift

Triangle B is to the left of triangle D. To move from B to D, we move right. Δx = + 3.
The translation vector is \((3, - 3)\).

Part (e): Map C onto D

• Step 1: Determine vertical shift

Triangle C is above triangle D. To move from C to D, we move down. Δy = - 1 (assuming the vertical distance is 1 unit? Wait, re - evaluating: Wait, looking at the grid, C and D: Let's check the vertical and horizontal. Wait, C is to the left of D and above? Wait, no, let's re - check. Wait, maybe my initial count was wrong. Wait, let's take a vertex of C and D. Let's say the top - left vertex of C and D. For C to D: horizontal shift: D is 6 units to the right of C? Wait, no, maybe the grid is such that from C to D: horizontal shift (right) is 6? Wait, no, the original figure: Let's assume each triangle has a top - left vertex. For A: top - left at (x1,y1), B: (x1 + 5,y1), C: (x1,y1 - 2), D: (x1+6,y1 - 3). Wait, maybe better to count the horizontal and vertical differences. Let's take the top - left corner of each triangle:

• A: (let's say) (2, 5)
• B: (7, 5)
• C: (2, 3)
• D: (8, 2)

So for C (2,3) to D (8,2):

• Horizontal shift: 8 - 2 = 6 (right, so + 6)
• Vertical shift: 2 - 3 = - 1 (down, so - 1)

So translation vector is \((6, - 1)\). Wait, maybe my initial grid assumption was wrong. Alternatively, maybe the horizontal shift from C to D is 6 units right and 1 unit down.

Part (f): Map D onto C

• Step 1: Determine vertical shift

Triangle D is below triangle C. To move from D to C, we move up. Δy = + 1.

• Step 2: Determine horizontal shift

Triangle D is to the right of triangle C. To move from D to C, we move left. Δx = - 6.
The translation vector is \((- 6, 1)\).

Final Answers:

a) \(\boldsymbol{(0, - 2)}\)
b) \(\boldsymbol{(0, 2)}\)
c) \(\boldsymbol{(- 3, 3)}\)
d) \(\boldsymbol{(3, - 3)}\)
e) \(\boldsymbol{(6, - 1)}\) (depending on grid interpretation, may vary slightly but the method is horizontal and vertical shift calculation)
f) \(\boldsymbol{(- 6, 1)}\)